N,S]) consisting of those homomorphisms which induce isomorphisms between the respective Tate

cohomology groups of [X.

Then the 2-th Hochschild

cohomology group of A with coefficients in X is defined by

Again, if w is not a coboundary, then there is no ample division algebra in C(G,[omega],R); if w is a coboundary, then the iso-classes of ample division algebras in C(G,[omega],R) with D = R or H are parametrized by the second

cohomology group [H.

We wish to construct equivariant Chern classes for Real bundles in equivariant

cohomology groups with integral coefficients and our main requirement is that one recovers the classical Chern classes by forgetting the [C.

An important connection between the

cohomology groups of finite dimensional CSL-algebras (with values in the enveloping matrix algebras) and the homology groups of certain simplicial complexes was presented recently by Kraus and Schack [18].

MacLane and Whitehead have proved in [MW] that equivalence classes of so called crossed modules are in bijection with the elements of the third

cohomology group [H.

2]), r < d, be the first nonzero reduced

cohomology group of M.

D])= -2 so that the last

cohomology group must have positive dimension.

Section 3 contains the proprieties of two-dimensional local field we need, such as, duality and the vanishing of the second

cohomology group.

Among the topics are genus change in inseparable extensions of functional fields, the homology of noetherian rings and local rings, the

cohomology groups of tori in infinite Galois extensions of number fields, an algorithm for determining the type of a singular fiber in an elliptic pencil, variation of the canonical height of a point depending on a parameter, the non-existence of certain Galois extensions of Q unramified outside two, and refining Gross' conjecture on the values of abelian L-functions.

Basic forms are preserved by the exterior derivative and are used to define basic de-Rham

cohomology groups [H*.

The endomorphism algebra of a tilting module preserves many significant invariants, for example, the center of an algebra, the number of nonisomorphic simple modules, the Hochschild

cohomology groups, and Cartan determinants.